Superfluidity of bosons on a deformable lattice

We study the superfluid properties of a system of interacting bosons on a lattice, which, moreover, are coupled to the vibrational modes of this lattice, treated here in terms of Einstein phonon modes. The ground state corresponds to two correlated condensates: that of the bosons and that of the phonons. Two competing effects determine the common collective sound-wave-like mode with sound velocity $v,$ arising from gauge symmetry breaking. (i) The sound velocity ${v}_{0}$ (corresponding to a weakly interacting Bose system on a rigid lattice) in the lowest-order approximation is reduced due to reduction of the repulsive boson-boson interaction, arising from the attractive part of the phonon-mediated interaction in the static limit. (ii) The second-order correction to the sound velocity is enhanced as compared to that of bosons on a rigid lattice when the boson-phonon interaction is switched on due to the retarded nature of the phonon-mediated interaction. The overall effect is that the sound velocity is essentially unaffected by the coupling with phonons, indicating the robustness of the superfluid state. The induction of a coherent state in the phonon system driven by the condensation of the bosons could be of experimental significance, permitting spectroscopic detection of superfluid properties of bosons. Our results are based on an extension of the Beliaev-Popov formalism for a weakly interacting Bose gas on a rigid lattice to one on a deformable lattice with which it interacts.